Weak error analysis for semilinear stochastic Volterra equations with additive noise

作者：Andersson Adam^*; Kovacs Mihaly; Larsson Stig

来源：Journal of Mathematical Analysis and Applications, 2016, 437(2): 1283-1304.

DOI：10.1016/j.jmaa.2015.09.016

• Stochastic Volterra equations
• Finite element method
• Backward Euler
• Convolution quadrature
• Strong and weak convergence
• Malliavin regularity

We prove a weak error estimate for the approximation in space and time of a semilinear stochastic Volterra integro-differential equation driven by additive space time Gaussian noise. We treat this equation in an abstract framework, in which parabolic stochastic partial differential equations are also included as a special case. The approximation in space is performed by a standard finite element method and in time by an implicit Euler method combined with a convolution quadrature. The weak rate of convergence is proved to be twice the strong rate, as expected. Our convergence result concerns not only functionals of the solution at a fixed time but also more complicated functionals of the entire path and includes convergence of covariances and higher order statistics. The proof does not rely on a Kolmogorov equation. Instead it is based on a duality argument from Malliavin calculus.

• 出版日期2016-5-15